The subject matter of the present invention relates to a software based method and associated apparatus for mapping uncertainty by producing one or more “maps”, such as a probability map or a cutoff map or a confidence limit map, and or one or more cubes, based on the conditional simulation of a set of random variables.
The estimation at any spatial location (x, y, z) of the value of a parameter (i.e., a random variable), such as porosity or permeability, from a set of scattered observations of data representing such a parameter may be achieved by a method known as “Kriging”. For a reference which describes “Kriging”, refer to either of the following two references: (1) Journe, A. G. “Fundamentals of Geostatistics in Five Lessons”, Short course in Geology, vol 8, 44 pp, AGU, Washington, D.C. 1989, or (2) Deutsch, Clayton V. and Andre Journel, “GSLIB Geostatistical Software Library and User's Guide second edition”, Oxford University Press, New York, Oxford, 1998; the disclosures in each of the above two references which discuss “Kriging” are incorporated by reference into this specification. For example, if a set of scattered data samples represents porosity (obtained, for example, by mapping a cross section of an earth formation through which a plurality of wellbores are drilled as indicated in FIGS. 4 and 5), when the cross section is gridded, the “Kriging” method can determine, at each intersection of the grid, the expected (mean) value of porosity and its standard deviation.
The result of estimating this value at a regular grid results in a smooth surface of the expected value generally following the data (see FIGS. 4 through 16). If the principle of exactitude is applied, this surface will actually go through the data if the data falls on a grid location. The error variance is also calculated at each grid location. At the data locations, this error variance equals zero unless the observation error variance is included in the calculation, in which case, this is the error variance at the data locations.
It is common practice to render a more realistic estimate of the variable by randomizing the answer using the estimated error variance. This is called a ‘conditional simulation’ because it is conditioned by the data and produces one out of any number of possible realizations. It is then customary to assess the risk associated with exploitation of the estimated accumulation of ore, hydrocarbons or other valuable commodities by sampling a good many of the realizations and ranking them in low, medium, and high ranges, according to the economic value of the estimated deposits. This is a lengthy and costly procedure.
Therefore, a new method of mapping uncertainty is needed.
A method of mapping an earth formation has been disclosed in U.S. Pat. Nos. 5,563,949 and 5,995,907. Another method for mapping an earth formation and generating a cube which contains a plurality of such maps is disclosed in prior pending application Ser. No. 09/377,573, filed Aug. 19, 1999, and entitled “Seismic signal processing method and apparatus for generating a cube of variance values”.